Research
Habilitation Thesis
- Uniform Error Estimation for Convection-Diffusion Problems, TU-Dresden, 2014,
mentor: Prof. H.-G. Roos,
arXiv:1403.0407
PhD Thesis
- Singularly perturbed problems with characteristic layers: Supercloseness and postprocessing, TU-Dresden, 2008,
supervisor: Jun.-Prof. T. Linß
Diploma Thesis
- Entwicklung eines impliziten LES Modells für kompressible Turbulenz, TU-Dresden, 2004,
supervisors: Prof. H.-G. Roos, Prof. N.A. Adams,
in German
Publications in journals
- S. Franz:
Anisotropic $H_{div}$-norm error estimates for rectangular $H_{div}$-elements,
Appl. Math. Letters 121 (2021), 107453 - S. Franz:
Continuous time integration for changing time systems. - S. Franz, H.-G. Roos:
Error estimates in balanced norms of finite element methods for higher order reaction-diffusion problems.
Int. J. Numer. Anal. Model., 17 (2020), 532–542 - M. Brdar, S. Franz, H.-G. Roos:
Numerical treatment for singularly perturbed fourth-order two-parameter problems,
ETNA 51 (2019), 50-62 - P. Constantinou, S. Franz, L. Ludwig, Chr. Xenophontos:
A mixed hp FEM for the approximation of 4th order singularly perturbed problems on smooth domains
Numer. Methods Partial Differential Equations, 35 (2019) 1, 114-127 - S. Franz, S. Trostorff, M. Waurick:
Numerical methods for changing type systems
IMA J. Numer. Anal., 39 (2) (2019), 1009-1038 - L. Teofanov, M. Brdar, S. Franz, H. Zarin:
SDFEM for a singularly perturbed problem with two parameters.
Calcolo (2018) - P. Constantinou, S. Franz, L. Ludwig, Chr. Xenophontos:
Finite element approximation of reaction-diffusion problems using an exponentially graded mesh
Computer Math. Appl. (2018), 76 (2018), 2523-2534 - S. Franz, M. Waurick:
Resolvent estimates and numerical implementation for the homogenisation of one-dimensional periodic mixed type problems
ZAMM, 98 (2018) 7, 1284-1294 - S. Franz, G. Matthies:
A Unified Framework for Time-Dependent Singularly Perturbed Problems with Discontinuous Galerkin in Time
Math. Comp., 87 (2018) 313, 2113-2132 - S. Franz, K. Höhne, G. Matthies:
Grad-Div stabilisation applied to inf-sup stable discretisations with S-type meshes for the Oseen problem
IMA J. Numer. Anal., 38 (1) (2018), 299-329 - S. Franz, K. Höhne, M. Waurick:
A solution decomposition for a singularly perturbed 4th order problem
Analysis, 37 (2017) 2, 77-100 - S. Franz, Chr. Xenophontos:
A Short Note on the connection between layer-adapted exponentially graded and S-type meshes
CMAM, 18 (2017) 2, 199-202 - S. Franz:
Convergence of LPS-FEM for convection-diffusion problems on layer-adapted meshes
BIT Numerical Mathematics, 57(3) (2017), 771-786 - S. Franz:
Analysis for a family of continuous-discontinuous Galerkin FEM for convection-diffusion problems.
Appl. Numer. Math., 110 (2016), 93-109 - S. Franz, L. Ludwig, Chr. Xenophontos:
Finite element approximation of convection-diffusion problems using an exponentially graded mesh
Computer Math. Appl., 72 (2016) 6, 1532-1540 - S. Franz, H.-G. Roos:
Robust error estimation in energy and balanced norms for singularly perturbed forth order problems
Computer Math. Appl., 72 (2016) 1, 233-247 - S. Franz, H.-G. Roos:
Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers.
Calcolo, 51 (2014) 3, 423-440 , Addendum - S. Franz, H.-G. Roos:
Superconvergence of a Galerkin FEM for Higher-Order Elements in Convection-Diffusion Problems.
Numer. Math. Theor. Meth. Appl., 7 (2014) 3, 356-373 , arXiv:1307.7543 - S. Franz, H.-G. Roos, A. Wachtel:
A C0 interior penalty method for a singularly-perturbed fourth-order elliptic problem on a layer-adapted mesh.
Numer. Methods Partial Differential Equations, 30 (2014) 3, 838-861 - S. Franz:
Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems.
Appl. Numer. Math., 76 (2014), 132-144 , arXiv:1304.7443 - S. Franz:
Convergence Phenomena of Q_p-Elements for Convection-Diffusion Problems.
Numer. Methods Partial Differential Equations, 29 (2013) 1, 280-296 - S. Franz, R. Gärtner, H.-G. Roos and A. Voigt:
A Note on the Convergence Analysis of a Diffuse-Domain Approach.
CMAM, 12 (2012) 2, 153-167 - S. Franz, R. B. Kellogg, M. Stynes:
Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities.
Math. Comp., 81 (2012) 278, 661-685 - S. Franz, N. Kopteva:
Green's function estimates for a singularly perturbed convection-diffusion problem.
Journal of Differential Equations, 252 (2012), 1521-1545 - S. Franz, N. Kopteva:
Green's function estimates for a singularly perturbed convection-diffusion problem in three dimensions.
Int. J. Numer. Anal. Model. Ser. B, 2 (2011) 2-3, 124-141 - S. Franz, G. Matthies:
Convergence on Layer-adapted Meshes and Anisotropic Interpolation Error Estimates of Non-Standard Higher Order Finite Elements.
Appl. Numer. Math., 61 (2011), 723-737 , corrected preprint - S. Franz:
SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers.
BIT Numerical Mathematics, 51 (2011) 3, 631-651 - S. Franz, H.-G. Roos:
The Capriciousness of Numerical Methods for Singular Perturbations.
SIAM Review, 53 (2011) 1, 157-173 - S. Franz, G. Matthies:
Local projection stabilisation on S-type meshes for convection-diffusion problems with characteristic layers.
Computing, 87 (2010) 3-4, 135-167 - S. Franz, T. Linß, H.-G. Roos, S. Schiller:
Uniform superconvergence of a finite element method with edge stabilization for convection-diffusion problems.
J. Comp. Math., 28 (2010) 1, 32-44 - S. Franz, L. Tobiska, H. Zarin:
A New Approach to Recovery of Discontinuous Galerkin.
J. Comp. Math., 27 (2009) 6, 697-712 - S. Franz, F. Liu, H.-G. Roos, M. Stynes, A. Zhou:
The combination technique for a two-dimensional convection-diffusion problem with exponential layers.
Appl. Math., 54 (2009) 3, 203-223 - S. Franz, T. Linß, H.-G. Roos:
Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers,
Appl. Numer. Math., 58 (2008) 12, 1818-1829 - S. Franz:
Continuous interior penalty method on a Shishkin mesh for convection-diffusion problems with characteristic boundary layers,
Comput. Meth. Appl. Mech. Engng., 197 (2008) 45-48, 3679-3686 - S. Franz, T. Linß:
Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problem with characteristic layers,
Numer. Methods Partial Differential Equations, 24 (2008) 1, 144-164 - N.A. Adams, S. Hickel, S. Franz:
Implicit subgrid-scale modeling by adaptive deconvolution,
J. Comput. Phys. 200 (2004) 2, 412-431
Publications in proceedings
- J. Orwat, S. Franz:
Average graph of land surface curvatures induced mining operation determined by use of fourth order polynomial.
In AIP Conference Proceedings 2293 (2020), 1, 350009. - S. Franz, M. Waurick:
Homogenisation of parabolic/hyperbolic media.
In Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2018,
Lecture Notes in Computational Science and Engineering 135 (2020), 215–229, Springer, Berlin Heidelberg - S. Franz, H.-G. Roos:
On robust error estimation for singularly perturbed fourth-order problems
In Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016
Lecture Notes in Computational Science and Engineering 120 (2017), 77-85, Springer, Berlin Heidelberg - S. Franz, H.-G. Roos:
Robust error estimation in energy and balanced norms for singularly perturbed forth order problems
Model. Anal. Inform. Sist., 23:3 (2016), 364-369 - S. Franz, J. Rubisch:
A discontinuous Galerkin method for fractional differential equations
PAMM, 16 (2016), 761-762 - S. Franz, N. Kopteva:
On the Sharpness of Green's function estimates for a convection-diffusion problem.
In Proceedings of the Fifth Conference on Finite Difference Methods: Theory and Applications (FDM: T&A 2010), (2011), 44-57, Rousse University Press, arXiv:1102.4520v2 - S. Franz, R. B. Kellogg, M. Stynes:
On the choice of mesh for a singularly perturbed problem with a corner singularity.
In BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods
Lecture Notes in Computational Science and Engineering 81 (2010), 119-126, Springer, Berlin Heidelberg - S. Franz, G. Matthies:
Local projection stabilisation on layer-adapted meshes for convection-diffusion problems with characteristic layers (Part I and II)
In BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods
Lecture Notes in Computational Science and Engineering 81 (2010), 127-138, Springer, Berlin Heidelberg - S. Franz, T. Linß:
Superconvergence analysis of Galerkin FEM and SDFEM for Elliptic Problems with characteristic layers,
Proceedings in Applied Mathematics and Mechanics 6 (2006) 1, 737-738. - N.A. Adams, S. Franz, S. Hickel, P. Koumoutsakos:
Optimization of an implicit subgrid-scale model for LES,
In Proceedings of the 21st International Congress of Theoretical and Applied Mechanics, Warsaw, Poland, 2004
Preprints and submitted papers
- S. Franz:
Singularly perturbed reaction-diffusion problems as first order systems,
submitted, arxiv:2103.10750 - Chr. Xenophontos, S. Franz, E. Sykopetritou :
Mixed hp FEM for singularly perturbed fourth order boundary value problems with two small parameters
submitted, arXiv:2008.01804 - S. Franz, N. Kopteva:
Full analysis of the Green's function for a singularly perturbed convection-diffusion problem in three dimensions.
arXiv:1103.2948v2 - S. Franz:
A recovery operator for the combination method.
Preprint MATH-NM-09-2008, 10/2008, preprint